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Bayes' Rule is a theorem that allows us to turn around a conditional probability statement. This rule allows you to update or revise your probabilities in light of new information. Essentially, it reverses the conditional probability formula, allowing you to find the probability of A given B from the probability of B given A.
Suppose there's a famous television show called "Go For Broke.” On the show, a person gets to roll a die to determine which jar they're going to draw a colored chip from. If the chip is green, they will win a prize, and if the chip is red, they won’t win anything.
Roll a 1 or 2: Pick from Jar A |
Roll a 3, 4, 5, or 6: Pick from Jar B |
Rolling a 1 or a 2 and being able to select from jar A gives the contestant a better probability of winning a prize. Here is the tree diagram of the probabilities in this game:
But recall that the "and" probability of two dependent events is the probability of one event times the conditional probability of the second event, given the first event occurred. So we can actually rewrite this equation:
When rewritten in this new formula, you may notice that the probability of G given A helps us to find the probability of A given G. When we actually go through and run the scenarios, the probability of A and G was one of the branches on the tree diagram, or 1/7.
To find the probability of G, we can see from the tree diagram that this can happen in two ways: selecting from jar A or jar B.
Plugging in this information, the equation simplifies down to the fraction 9/16::
What this means is that out of every 16 times you pick the green, 9 of those times it came from jar A. This is a surprising conclusion! The probability that you would pick from jar A, given that you ended up winning, is actually over half.
Another example of when you might use Bayes' theorem is during an actual army practice to find the wreckage of a plane.
Suppose that a plane was going from Miami to Mexico City, but then it crashed somewhere in the Gulf of Mexico. Here is a map of the scenario:
As the map shows, the plane may have lost radio contact in one location. Therefore, because they don't know what else might have happened with the plane, the search party might start searching in that area. However, perhaps a few days go by, and people find debris in a different location.
Bayes' theorem allows you to analyze the probability that the plane crashed in the region where they lost radio contact, given that the debris was found elsewhere, where the green circle is.
If that probability is low, you might choose to start searching in the green region, assuming that the plane tried to go around the storm by going north.
The idea is that we can update our guesses based on new information.
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